One of the most useful modulation techniques in applications requiring low receiver complexity is Frequency Shift Keying (FSK). In an FSK scheme, digital information is transmitted by changing the frequency of a carrier signal.

## Binary FSK

Binary FSK (BFSK) is the simplest form of FSK where the two bits 0 and 1 correspond to two distinct carrier frequencies $F_0$ and $F_1$ to be sent over the air. The bits can be translated into symbols through the relations

\begin{equation*}

\begin{aligned}

0 \quad & \rightarrow \quad -1 \\

1 \quad & \rightarrow \quad +1

\end{aligned}

\end{equation*}

This enables us to write the two frequencies $F_i$ with $i ~ \epsilon ~ \{0,1\}$ as

\begin{equation*}

F_i = F_c + (-1)^{i+1} \cdot \Delta_F = F_c \pm \Delta_F

\end{equation*}

where $F_c$ is the nominal carrier frequency and $\Delta_F$ is the peak frequency deviation from this carrier frequency. The signal waveform can now be written as

\begin{equation}\label{eqFSKwaveform}

s(t) = A \cos \left(2\pi F_i t +\phi \right)= A \cos\Big[2\pi \big(F_c \pm \Delta_F\big) t + \phi\Big]

\end{equation}

where $0\le t \le T_b$ and $\phi$ is an arbitrary phase. Figure below displays a BFSK waveform for a random stream of data at a rate of $R_b = 1/T_b$. Note that we are not distinguishing between a bit period and a symbol period because both are the same for a binary modulation technique.

Let us turn towards the mostly used block for FSK demodulation in GNU Radio.

## Quadrature Demod Block

As far as an FSK receiver is concerned, a commonly used method in GNU Radio implementations is the Quadrature Demod block. A block diagram of such a detector is reproduced below from GNU Radio wiki where the block Quadrature Demod is encircled in red.

Such a structure can be used to demodulate several frequency modulation schemes such as FM, FSK and GMSK. The input to the block is the complex baseband waveform. Within the block, a product of the one-sample delayed input and the conjugate original signal is computed, the output of which is a complex number.

\begin{equation*}

z(nT_S) = s(nT_S)\cdot s^*\left[(n-1)T_S\right]

\end{equation*}

The argument of this complex number is the signal frequency relative to the sample rate multiplied with the gain, as we will see later.

## Demodulating an FSK Signal

To see the role of a Quadrature Demod block in demodulating an FSK signal, let us consider a complex version of the FSK modulated signal described in Eq (\ref{eqFSKwaveform}) which just showed the real part.

\begin{equation*}

s(t) = A e^{j \left(2\pi F_i t + \phi\right)}

\end{equation*}

This signal is sampled at a rate of $F_S=1/T_S$ where $T_S$ is the sample period.

\begin{equation*}

s(nT_S) = A e^{ j \left(2\pi F_i nT_S + \phi\right)} = A e^{j \left(2\pi \frac {F_i}{F_S} n + \phi\right)}

\end{equation*}

### Generating the Waveform

In a very simple GNU Radio Companion flowgraph below, an FSK waveform is generated with center frequency 1.5 kHz and a frequency deviation of 500 Hz. This results in two carrier waves, one at 1 kHz representing a 0 and the other at 2 kHz representing a 1. This is implemented in the following manner.

- Multiply the bits, e.g., 0,1,1,0,0,1,…, with the higher frequency wave at 2 kHz that produces a 2 kHz wave for 1 bits and a blank space of zeros for 0 bits.
- Multiply one minus the bits, e.g., 1- (0,1,1,0,0,1,…), with the lower frequency wave at 1 kHz that produces a 1 kHz wave for 0 bits and a blank space of zeros for 1 bits.
- Add the two waves together thus generating a BFSK waveform.

### Frequency Xlating FIR Filter

At the Rx side, a frequency translating FIR filter with a center frequency of 1.5 kHz is employed. According to its documentation, this block performs a frequency translation on the signal, as well as downsamples the signal by running a decimating FIR filter on it. This operation places the modulated signal at baseband and hence the two possible frequencies are now located at $\pm 500$ Hz. This is shown by two impulses at $\pm 500$ Hz below that is the output of the flowgraph described above.

The command firdes.low_pass(1.0,samp_rate,900,300) uses a lowpass filter with unity gain, a sample rate of 32 kHz, a cutoff frequency of 900 Hz and a transition bandwidth of 300 Hz. This implies that the edge of passband lies at 900-300=600 Hz, well beyond the impulse location of 500 Hz. A decimation operation can also be used at this stage as an argument of Frequency Xlating FIR Filter to operate only at a minimum number of samples dictated by the sampling theorem. However I am not decimating the signal here since my objective is to display the proper waveforms for which a higher the number of samples produces better results.

### Operation of Quadrature Demod Block

Now the Quadrature Demod block computes the product of this signal with a unit delayed and conjugated version of itself.

\begin{equation*}

z(nT_S) = s(nT_S)\cdot s^*\left[(n-1)T_S\right]

\end{equation*}

This simplifies to the following expression.

\begin{equation*}

\begin{aligned}

z(nT_S) &= A e^{j \left(2\pi \frac{F_i}{F_S}n + \phi\right) }\cdot A e^{-j \left(2\pi \frac{F_i}{F_S}(n-1) + \phi\right) } \\

&= A^2 e^{j \left(2\pi \frac{F_i}{F_S}n + \phi – 2\pi \frac{F_i}{F_S}(n-1) – \phi\right) }\\

&= A^2 e^{j \left(2\pi \frac{F_i}{F_S}\right)}

\end{aligned}

\end{equation*}

Taking its argument reveals the underlying modulation frequency.

\begin{equation}\label{eqArg}

\mathrm{arg} \big\{z(nT_S)\big\} = 2\pi \frac{F_i}{F_S}

\end{equation}

Considering that $F_i$ is simply a sequence of two different frequencies depending on whether out input was a 0 or 1, the output of this operation is a square waveform similar to a Pulse Amplitude Waveform (PAM) with a modulation sequence of $\pm 2\pi/F_S$ (corresponding to $\pm 1$). This is the important point. The binary frequency modulated waveform has been sort of converted into a pulse amplitude waveform due to this operation. This stream of -1s and 1s can be detected in the Quadrature Demod Output shown in the figure above. This stream can be passed through a binary slicer to produce the final bit stream.

### Timing Synchronization

Any of the well known timing recovery algorithms can be applied for synchornization purpose. At first sight, it does not make sense that algorithms originally devised for a linear modulation scheme are applicable to an FSK waveform. Indeed, this is a great cause of confusion in SDR beginners. However, after the above explanation of demodulation process, we can see that a binary FSK waveform is indeed a square wave signal similar to a linearly modulated sequence. Therefore, we can apply any well known timing synchronization scheme such as a Gardner algorithm, polyphase clock sync, Mueller and Müller clock recovery, and so on.

A word of caution: GNU Radio is a tool to implement signal processing algorithms while leaving the underlying algorithms choice and development for custom blocks to the user. For a lab demo or a small project, there is no harm in performing the timing synchronization through any of the above mentioned techniques. On the other hand, a professional solution working under several different operating conditions should be more suited to the exact problem itself.

The problem here is that the argument function $\mathrm{arg}\{\cdot\}$ used in Eq (\ref{eqArg}) delivers the principal value of the argument which is highly non-linear and an unwrapping process would be needed. This becomes difficult as noise does not translate nicely from arguments to a linear representation and severely limits the performance at moderate to low SNRs.